We should probably start with Earth. Fairly obvious what it’s modeled after.
I believe that it makes its first appearance on the series in the episode "A Head in the Polls," in which the head of Richard
Nixon in a jar is elected president of Earth in a robot-vote lanslide. It also pops up in the episode "The Taste of Freedom",
which is, incidentally, largely about flag desecration. The second flag appears at the back of the Earth Legislature chamber
in the episode "When Aliens Attack." The vertical-hoist equivalent of the regular flag, I suppose.

DOOP

DOOP

And now onto the DOOP. Yes, the DOOP. Stop snickering, dammit; it stands for ‘Democratic Order
of Planets.’ The interstellar government of the Futurama universe, and quite an amusing satire of Star Trek’s
Federation, just as DOOP’s ace space captain Zapp Brannigan is basically a bombastic, horny(er) Kirk clone. The
flag is basically in the UN style, with DOOP centered in lowercase in a deliberately retro-70s font in front of a symbol.
Though the field is a deeper blue, and I believe that those are supposed to be laurel branches, not olive. The symbol
over which 'DOOP' is superimposed bears a striking resemblance to that of the UN International Atomic Energy Agency, by the
way. Look it up.

DOOP Arms

DOOP Badge/Roundel

There is also a coat of arms. It first appears in the episode "Brannigan Begin Again," as does the flag. The only difference
from the symbol on the flag is that the spheres in the elliptical orbits are a) larger and b) golden.

There's a roundel/badge too. It shows up fairly often--every episode that features the Nimbus, and on Fry's helmet in "When
Aliens Attack.". A simplified, unicolored version of the logo, without the wreath or the spheres in the elliptical orbits.

THE NEUTRAL PLANET

Neutral Planet

As seen in the episode where Zapp is drummed out of the military and becomes a Planet Express employee. (Capital: Neutropolis
Motto: Live Free Or Don't). The flag can be seen on embassy row in "The Taste of Freedom". Though it goes by
rather quickly so I'm not sure if it's closer to beige or gray...

Update: I gather it's gray. It figures the neutrals wouldn't use as wild a color as beige on their flag.

Second Update: Ok, there's a closeup in the president's office in "Brannigan Begin Again." Ready for the final
word on the topic? It's... neither! Kind of a indescribable grayish-beige color. Damn neutrals and
their neutrality.

SPACE PIRATES

Space Pirates

The Space Pirates, play a part in the opening scenes of episode "Godfellas." Points of interest include a three-eyed
skull, odd curved-pennant shape and the choice of maroon instead of black, presumably for enhanced visibility in space.

DECAPOD 10

Decapod 10 is the homeworld of comedy-relief space lobster Dr. Zoidberg, and a rather vexxillogenous world, it would
seem, as we get four flags in only two appearances.
Two flags show up in "Why Must I Be a Crustacean In Love:"
1. A long, narrow pink flag, with the lower fourth taken up by blue waves, cresting from left to right. Appears above the
welcome booth at the Decapod 10 spaceport. I think this might be the flag of the Mating Frenzy.
2. A vertical, dark-purple swallowtail with a thin light-purple border, visible behind the Decapodian elder in the arena
while he presides over Fry & Zoidberg's fight to the death.

DECAPOD 10

DECAPODIAN MATING FRENZY (?)

DECAPOD 10, AGAIN

Two further Decapodian flags pop up in the episode "The Taste of Freedom," in which the Decapodians join the club and
invade Earth (Hey, eveyone else is doing it!). #3 is visible in the ambassador's office, and then later #4 can be seen
flying atop the Mobile Opression Palace.

DECAPODIAN EARTH OPPRESSION FORCES

The former is a rather festive design with twin, narrow purple stripes on a violet background; the latter is likely the flag
of the Decapodian occupation government of Earth--looks like an attempt to combine Nazi and Soviet symbols, with a white disk
on a dark red field bearing a hammer in the grasp of a Decapodian claw.

Copyright 20th Century Fox, 30th Century Fox, the head of Rupert Murdoch in a jar